Results 1 to 5 of 5

Math Help - Prime counting function

  1. #1
    MHF Contributor
    Joined
    Oct 2005
    From
    Earth
    Posts
    1,599

    Prime counting function

    I'm trying to spark some interest in discussion in prime number theory and the Riemann Zeta Function/Hypothesis. The prime counting function, \pi(x), outputs the number of primes less than or equal to x. Gauss suggested at 15 that a good approximation for this prime counting function was Li(x), where Li(x)=\int_{0}^{\infty}\frac{dx}{\log(x)} (I realize that log(1)=0). This is a good approximation and although it seems to always bound the prime counting function, they do eventually cross.

    Anyway, here is an explicit formula for a prime counting function.

    J(x)=Li(x)-\sum_{\rho}\frac{x^{\rho}}{\rho}-\ln(2)+\int_{x}^{\infty}\frac{dt}{t(t^2-1}\ln(t)}

    Here \rho represents the non-trivial zeros of the Riemann Zeta Function in the critical strip. Since there are infinitely many zeros in the critical strip this formula is not feasible to use for actual caculations, but my question is that if we could use this formula would it output integers? I mean would it say exactly how many primes are less than or equal to x?

    I'd appreciate any other thoughts to add to my own.
    Last edited by Jameson; June 11th 2006 at 03:28 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by Jameson
    I'm trying to spark some interest in discussion in prime number theory and the Riemann Zeta Function/Hypothesis.
    Good luck with that.

    Quote Originally Posted by Jameson
    where Li(x)=\int_{0}^{\infty}\frac{1}{\log(x)}
    Perhaps you mean to say,
    \mbox{Li}(x)=\int^x_2\frac{du}{\log u}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Speaking about the prime counting function there is an elemantary proof (using only basic analysis) in my book that
    \lim_{n\to \infty}\frac{\pi (n)}{n}=0
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Oct 2005
    From
    Earth
    Posts
    1,599
    I forgot the dx, but I meant what I wrote. I technically should use limit notation to avoid the asymptote in the function, but I assume that you can notice that. There are two common Li(x) functions it seems.

    http://en.wikipedia.org/wiki/Logarithmic_integral
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Oct 2005
    From
    Earth
    Posts
    1,599
    Quote Originally Posted by ThePerfectHacker
    Speaking about the prime counting function there is an elemantary proof (using only basic analysis) in my book that
    \lim_{n\to \infty}\frac{\pi (n)}{n}=0
    This is interesting. I'll look into it.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Riemann's Prime Counting Function
    Posted in the Math Challenge Problems Forum
    Replies: 1
    Last Post: July 7th 2010, 11:51 AM
  2. help me with counting of function
    Posted in the Discrete Math Forum
    Replies: 10
    Last Post: October 26th 2009, 06:36 PM
  3. Prime-counting function
    Posted in the Math Challenge Problems Forum
    Replies: 2
    Last Post: October 25th 2009, 01:10 AM
  4. Riemann's prime counting function
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: April 8th 2009, 05:31 PM
  5. Inequality with the prime-counting function
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 17th 2008, 12:07 PM

Search Tags


/mathhelpforum @mathhelpforum